A semigroup approach to fractional powers

We say that A has fractional powers {A _t } _t≥0 if there exists a nondegenerate C-regularized semigroup {W(t)} _t≥0 such that A=C ⁻¹ W(1); then A _t ≡C ⁻¹ W(t). We show that this generalizes the usual definition of fractional powers for nonnegative operators, and enables many operators with spectrum containing the entire unit disc to have fractional powers. Our definition gives clear, simple proofs of the basic properties of fractional powers. We show that, for nonnegative operators, the fractional powers with the property that, if A is of type θ, then A _t is of type t θ, whenever t θ<π, are unique. More generally, for injective G∈B(X) commuting with A, we show that an operator A of G-regularized type θ has a unique family of fractional powers with the property that A _t is of G-regularized type t θ whenever t θ<π. This leads to a construction of fractional powers of operators with polynomially bounded resolvent outside of an appropriate sector. We show that an operator is of regularized type if and only if it has exponentially bounded regularized imaginary powers. This work was done while the second author was visiting Ohio University, with funding from Universitat de València. He would like to thank Ohio University and Professor deLaubenfels for their hospitality and support.     

Variational Inequalities in Hilbert Spaces with Measures and Optimal Stopping Problems

We study the existence theory for parabolic variational inequalities in weighted L ² spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L ² setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coefficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.     

A weighted Lq‐approach to Oseen flow around a rotating body

We study time-periodic Oseen flows past a rotating body in ℝ³ proving weighted a priori estimates in L^q-spaces using Muckenhoupt weights. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms (ω ∧ x) ⋅ ∇ u and −ω ∧ u in the equation of momentum where ω denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood–Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones' factorization theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

On the Navier–Stokes equations with free convection in three‐dimensional unbounded triangular channels

The quaternionic calculus is a powerful tool for treating the Navier–Stokes equations very elegantly and in a compact form, through the evaluation of two types of integral operators: the Teodorescu operator and the quaternionic Bergman projector. While the integral kernel of the Teodorescu transform is universal for all domains, the kernel function of the Bergman projector, called the Bergman kernel, depends on the geometry of the domain. In this paper, we use special variants of quaternionic‐holomorphic multiperiodic functions in order to obtain explicit formulas for unbounded three‐dimensional parallel plate channels, rectangular block domains and regular triangular channels. Copyright © 2007 John Wiley & Sons, Ltd.     

Multilinear Singular and Fractional Integrals

In this paper, we treat a class of non-standard commutators with higher order remainders in the Lipschitz spaces and give （L^v, L^q）, （H^p, L^q） boundedness and the boundedness in the Triebel- Lizorkin spaces. Our results give simplified proofs of the recent works by Chen, and extend his result.     

Sturm-Liouville operators with indefinite weight functions and eigenvalue depending boundary conditions

We consider a class of boundary value problems for Sturm-Liouville operators with indefinite weight functions. The spectral parameter appears nonlinearly in the boundary condition in the form of a function τ which has the property that λ?λτ(λ) is a generalized Nevanlinna function. We construct linearizations of these boundary value problems and study their spectral properties.     

Scalar Derivatives in Hilbert Spaces

Computational formulae for scalar derivatives of mappings and pairs of mappings in the direction of a set will be given. These formulae are very important for explicitly verifying conditions of existence for fixed point theorems, surjectivity theorems, integral equations, variational inequalities and complementarity problems under additional differentiability conditions. To emphasize this idea at the end of the paper we give an application to complementarity problems. Some theorems which extend the correspondence between monotone mappings and scalar derivatives from Euclidean spaces to Hilbert spaces will also be given.     

On Spencer Cohomologies of Affine Algebraic Varieties

This paper contains proofs and examples for some key facts about algebraic Spencer cohomologies preannounced elsewhere. The author hopes that here he has managed to build a more systematic discourse of the issues raised in earlier works.     

The Approximation Properties of Certain Interpolation Operators of Entire Exponential Type （Ⅱ）

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Wiener's lemma for twisted convolution and Gabor frames

We prove non-commutative versions of Wiener's Lemma on absolutely convergent Fourier series (a) for the case of twisted convolution and (b) for rotation algebras. As an application we solve some open problems about Gabor frames, among them the problem of Feichtinger and Janssen that is known in the literature as the ``irrational case'.